A193026 Decimal expansion of the constant term of the reduction of e^(-x) by x^2->x+1.

1, 3, 9, 7, 2, 9, 6, 5, 1, 6, 5, 0, 0, 0, 4, 4, 1, 5, 8, 0, 9, 3, 3, 4, 9, 3, 2, 3, 9, 0, 8, 9, 9, 4, 8, 6, 0, 5, 2, 6, 4, 0, 8, 7, 4, 3, 7, 2, 3, 7, 0, 9, 2, 3, 3, 5, 6, 4, 0, 8, 2, 8, 9, 0, 2, 5, 9, 3, 6, 7, 5, 9, 2, 4, 7, 1, 6, 5, 8, 7, 6, 7, 5, 3, 6, 4, 1, 3, 7, 5, 5, 7, 8, 3, 4, 4, 0, 2, 4, 3

Offset: 1

Author

Clark Kimberling, Jul 14 2011

Keywords

• nonn
• cons

Comments

Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010. The coefficient of x in this reduction is the constant at A099935.

Examples

			1.39729651650004415809334932390899486052640...
		

Crossrefs

Cf. A000045, A193010, A192232, A099935.

Programs

Mathematica

f[x_] := Exp[-x]; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
RealDigits[u0, 10]

Formula

From Amiram Eldar, Jan 18 2022: (Start)

Equals 1 + Sum_{k>=1} (-1)^k*Fibonacci(k-1)/k!.

Equals exp(-1/2)*(1 + sqrt(5)/5 + 2/(exp(sqrt(5))-1))*sinh(sqrt(5)/2). (End)